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Series: High Dimensional Seminar

TBANote the special time!

Series: School of Mathematics Colloquium

TBA

Series: Analysis Seminar

TBA

Series: High Dimensional Seminar

TBA

Series: High Dimensional Seminar

It has been known that when an equiangular tight frame (ETF) of size |Φ|=N exists, Φ ⊂ Fd (real or complex), for p > 2 the p-frame potential ∑i ≠ j | < φj, φk > |p achieves its minimum value on an ETF over all N sized collections of vectors. We are interested in minimizing a related quantity: 1/ N2 ∑i, j=1 | < φj, φk > |p . In particular we ask when there exists a configuration of vectors for which this quantity is minimized over all sized subsets of the real or complex sphere of a fixed dimension. Also of interest is the structure of minimizers over all unit vector subsets of Fd of size N. We shall present some results for p in (2, 4) along with numerical results and conjectures. Portions of this talk are based on recent work of D. Bilyk, A. Glazyrin, R. Matzke, and O. Vlasiuk.

Series: Stochastics Seminar

Series: Dissertation Defense

We model and analyze the dynamics of religious group membership and size. A groups
is distinguished by its strictness, which determines how much time group members are
expected to spend contributing to the group. Individuals differ in their rate of return for
time spent outside of their religious group. We construct a utility function that individ-
uals attempt to maximize, then find a Nash Equilibrium for religious group participation
with a heterogeneous population. We then model dynamics of group size by including
birth, death, and switching of individuals between groups. Group switching depends on
the strictness preferences of individuals and their probability of encountering members of
other groups. We show that in the case of only two groups one with finite strictness and
the other with zero there is a clear parameter combination that determines whether the
non-zero strictness group can survive over time, which is more difficult at higher strictness
levels. At the same time, we show that a higher than average birthrate can allow even the
highest strictness groups to survive. We also study the dynamics of several groups, gaining
insight into strategic choices of strictness values and displaying the rich behavior of the
model. We then move to the simultaneous-move two-group game where groups can set up
their strictnesses strategically to optimize the goals of the group. Affiliations are assumed
to have three types and each type of group has its own group utility function. Analysis
on the utility functions and Nash equilibria presents different behaviors of various types
of groups. Finally, we numerically simulated the process of new groups entering the reli-
gious marketplace which can be viewed as a sequence of Stackelberg games. Simulation
results show how the different types of religious groups distinguish themselves with regard
to strictness.

Series: Dissertation Defense

The Jacobian of a graph, also known as the sandpile group or the critical group, is a finite group abelian group associated to the graph; it has been independently discovered and studied by researchers from various areas. By the Matrix-Tree Theorem, the cardinality of the Jacobian is equal to the number of spanning trees of a graph. In this dissertation, we study several topics centered on a new family of bijections, named the geometric bijections, between the Jacobian and the set of spanning trees. An important feature of geometric bijections is that they are closely related to polyhedral geometry and the theory of oriented matroids despite their combinatorial description; in particular, they can be generalized to Jacobians of regular matroids, in which many previous works on Jacobians failed to generalize due to the lack of the notion of vertices.

Series: Dissertation Defense

The study of the longest common subsequences (LCSs) of two random words is a classical problem in computer science and bioinformatics. A problem of particular probabilistic interest is to determine the limiting behavior of the expectation and variance of the length of the LCS as the length of the random words grows without bounds. This dissertation studies the problem using both Monte-Carlo simulation and theoretical analysis. The specific problems studied include estimating the growth order of the variance, LCS based hypothesis testing method for sequences similarity, theoretical upper bounds for the Chv\'atal-Sankoff constant of multiple sequences, and theoretical growth order of the variance when the two random words have asymmetric distributions.

Series: Dissertation Defense

We will present three results in percolation and sequence analysis. In the first part, we will briefly show an exponential concentration inequality for transversal fluctuation of directed last passage site percolation. In the the second part, we will dive into the power lower bounds for all the r-th central moments ($r\ge1$) of the last passage time of directed site perolcation on a thin box. In the last part, we will partially answer a conjecture raised by Bukh and Zhou that the minimal expected length of the longest common subsequences between two i.i.d. random permutations with arbitrary distribution on the symmetric group is obtained when the distribution is uniform and thus lower bounded by $c\sqrt{n}$ by showing that some distribution can be iteratively constructed such that it gives strictly smaller expectation than uniform distribution and a quick cubic root of $n$ lower bound will also be shown.